Theorem of the highest weight

In representation theory, a branch of mathematics, the theorem of the highest weight classifies the irreducible representations of a complex semisimple Lie algebra .[1][2] There is a closely related theorem classifying the irreducible representations of a connected compact Lie group .[3] The theorem states that there is a bijection

from the set of "dominant integral elements" to the set of equivalence classes of irreducible representations of or . The difference between the two results is in the precise notion of "integral" in the definition of a dominant integral element. If is simply connected, this distinction disappears.

The theorem was originally proved by Élie Cartan in his 1913 paper.[4] The version of the theorem for a compact Lie group is due to Hermann Weyl. The theorem is one of the key pieces of representation theory of semisimple Lie algebras.

Statement

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Lie algebra case

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Let   be a finite-dimensional semisimple complex Lie algebra with Cartan subalgebra  . Let   be the associated root system. We then say that an element   is integral[5] if

 

is an integer for each root  . Next, we choose a set   of positive roots and we say that an element   is dominant if   for all  . An element   dominant integral if it is both dominant and integral. Finally, if   and   are in  , we say that   is higher[6] than   if   is expressible as a linear combination of positive roots with non-negative real coefficients.

A weight   of a representation   of   is then called a highest weight if   is higher than every other weight   of  .

The theorem of the highest weight then states:[2]

  • If   is a finite-dimensional irreducible representation of  , then   has a unique highest weight, and this highest weight is dominant integral.
  • If two finite-dimensional irreducible representations have the same highest weight, they are isomorphic.
  • For each dominant integral element  , there exists a finite-dimensional irreducible representation with highest weight  .

The most difficult part is the last one; the construction of a finite-dimensional irreducible representation with a prescribed highest weight.

The compact group case

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Let   be a connected compact Lie group with Lie algebra   and let   be the complexification of  . Let   be a maximal torus in   with Lie algebra  . Then   is a Cartan subalgebra of  , and we may form the associated root system  . The theory then proceeds in much the same way as in the Lie algebra case, with one crucial difference: the notion of integrality is different. Specifically, we say that an element   is analytically integral[7] if

 

is an integer whenever

 

where   is the identity element of  . Every analytically integral element is integral in the Lie algebra sense,[8] but there may be integral elements in the Lie algebra sense that are not analytically integral. This distinction reflects the fact that if   is not simply connected, there may be representations of   that do not come from representations of  . On the other hand, if   is simply connected, the notions of "integral" and "analytically integral" coincide.[3]

The theorem of the highest weight for representations of  [9] is then the same as in the Lie algebra case, except that "integral" is replaced by "analytically integral."

Proofs

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There are at least four proofs:

  • Hermann Weyl's original proof from the compact group point of view,[10] based on the Weyl character formula and the Peter–Weyl theorem.
  • The theory of Verma modules contains the highest weight theorem. This is the approach taken in many standard textbooks (e.g., Humphreys and Part II of Hall).
  • The Borel–Weil–Bott theorem constructs an irreducible representation as the space of global sections of an ample line bundle; the highest weight theorem results as a consequence. (The approach uses a fair bit of algebraic geometry but yields a very quick proof.)
  • The invariant theoretic approach: one constructs irreducible representations as subrepresentations of a tensor power of the standard representations. This approach is essentially due to H. Weyl and works quite well for classical groups.

See also

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Notes

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  1. ^ Dixmier 1996, Theorem 7.2.6.
  2. ^ a b Hall 2015 Theorems 9.4 and 9.5
  3. ^ a b Hall 2015 Theorem 12.6
  4. ^ Knapp, A. W. (2003). "Reviewed work: Matrix Groups: An Introduction to Lie Group Theory, Andrew Baker; Lie Groups: An Introduction through Linear Groups, Wulf Rossmann". The American Mathematical Monthly. 110 (5): 446–455. doi:10.2307/3647845. JSTOR 3647845.
  5. ^ Hall 2015 Section 8.7
  6. ^ Hall 2015 Section 8.8
  7. ^ Hall 2015 Definition 12.4
  8. ^ Hall 2015 Proposition 12.7
  9. ^ Hall 2015 Corollary 13.20
  10. ^ Hall 2015 Chapter 12

References

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